3 Description

The right eigenvector x, and the left eigenvector y, corresponding to an eigenvalue λ, are defined by:

Hx=λx and yHH=λyH or HTy=λ-y.

Note that even though H is real, λ, x and y may be complex. If x is an eigenvector corresponding to a complex eigenvalue λ, then the complex conjugate vector x- is the eigenvector corresponding to the complex conjugate eigenvalue λ-.

The eigenvectors are computed by inverse iteration. They are scaled so that, for a real eigenvector x,
maxxi=1,
and for a complex eigenvector,
max⁡Rexi+Im⁡xi=1.

If H has been formed by reduction of a real general matrix A to upper Hessenberg form, then the eigenvectors of H may be transformed to eigenvectors of A by a call to F08NGF (DORMHR).

4 References

5 Parameters

On entry: indicates whether left and/or right eigenvectors are to be computed.

JOB='R'

Only right eigenvectors are computed.

JOB='L'

Only left eigenvectors are computed.

JOB='B'

Both left and right eigenvectors are computed.

Constraint:
JOB='R', 'L' or 'B'.

2: EIGSRC – CHARACTER(1)Input

On entry: indicates whether the eigenvalues of H (stored in WR and WI) were found using F08PEF (DHSEQR).

EIGSRC='Q'

The eigenvalues of H were found using F08PEF (DHSEQR); thus if H has any zero subdiagonal elements (and so is block triangular), then the jth eigenvalue can be assumed to be an eigenvalue of the block containing the jth row/column. This property allows the routine to perform inverse iteration on just one diagonal block.

EIGSRC='N'

No such assumption is made and the routine performs inverse iteration using the whole matrix.

Constraint:
EIGSRC='Q' or 'N'.

3: INITV – CHARACTER(1)Input

On entry: indicates whether you are supplying initial estimates for the selected eigenvectors.

On entry: specifies which eigenvectors are to be computed. To obtain the real eigenvector corresponding to the real eigenvalue WRj, SELECTj must be set .TRUE.. To select the complex eigenvector corresponding to the complex eigenvalue WRj,WIj with complex conjugate (WRj+1,WIj+1), SELECTj and/or SELECTj+1 must be set .TRUE.; the eigenvector corresponding to the first eigenvalue in the pair is computed.

On exit: if a complex eigenvector was selected as specified above, then SELECTj is set to .TRUE. and SELECTj+1 to .FALSE..

On entry: the real and imaginary parts, respectively, of the eigenvalues of the matrix H. Complex conjugate pairs of values must be stored in consecutive elements of the arrays. If EIGSRC='Q', the arrays must be exactly as returned by F08PEF (DHSEQR).

On exit: some elements of WR may be modified, as close eigenvalues are perturbed slightly in searching for independent eigenvectors.

Note: the second dimension of the array VL
must be at least
max1,MM if JOB='L' or 'B' and at least 1 if JOB='R'.

On entry: if INITV='U' and JOB='L' or 'B', VL must contain starting vectors for inverse iteration for the left eigenvectors. Each starting vector must be stored in the same column or columns as will be used to store the corresponding eigenvector (see below).

On exit: if JOB='L' or 'B', VL contains the computed left eigenvectors (as specified by SELECT). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues. Corresponding to each selected real eigenvalue is a real eigenvector, occupying one column. Corresponding to each selected complex eigenvalue is a complex eigenvector, occupying two columns: the first column holds the real part and the second column holds the imaginary part.

Note: the second dimension of the array VR
must be at least
max1,MM if JOB='R' or 'B' and at least 1 if JOB='L'.

On entry: if INITV='U' and JOB='R' or 'B', VR must contain starting vectors for inverse iteration for the right eigenvectors. Each starting vector must be stored in the same column or columns as will be used to store the corresponding eigenvector (see below).

On exit: if JOB='R' or 'B', VR contains the computed right eigenvectors (as specified by SELECT). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues. Corresponding to each selected real eigenvalue is a real eigenvector, occupying one column. Corresponding to each selected complex eigenvalue is a complex eigenvector, occupying two columns: the first column holds the real part and the second column holds the imaginary part.

On entry: the first dimension of the array VR as declared in the (sub)program from which F08PKF (DHSEIN) is called.

Constraints:

if JOB='R' or 'B', LDVR≥max1,N;

if JOB='L', LDVR≥1.

14: MM – INTEGERInput

On entry: the number of columns in the arrays VL and/or VR . The actual number of columns required, m, is obtained by counting 1 for each selected real eigenvector and 2 for each selected complex eigenvector (see SELECT); 0≤m≤n.

Constraint:
MM≥m.

15: M – INTEGEROutput

On exit: m, the number of columns of VL and/or VR required to store the selected eigenvectors.

16: WORK(N+2×N) – REAL (KIND=nag_wp) arrayWorkspace

17: IFAILL(*) – INTEGER arrayOutput

Note: the dimension of the array IFAILL
must be at least
max1,MM if JOB='L' or 'B' and at least 1 if JOB='R'.

On exit: if JOB='L' or 'B', then IFAILLi=0 if the selected left eigenvector converged and IFAILLi=j>0 if the eigenvector stored in the ith column of VL (corresponding to the jth eigenvalue as held in WRj,WIj failed to converge. If the ith and i+1th columns of VL contain a selected complex eigenvector, then IFAILLi and IFAILLi+1 are set to the same value.

Note: the dimension of the array IFAILR
must be at least
max1,MM if JOB='R' or 'B' and at least 1 if JOB='L'.

On exit: if JOB='R' or 'B', then IFAILRi=0 if the selected right eigenvector converged and IFAILRi=j>0 if the eigenvector stored in the ith row or column of VR (corresponding to the jth eigenvalue as held in WRj,WIj) failed to converge. If the ith and i+1th rows or columns of VR contain a selected complex eigenvector, then IFAILRi and IFAILRi+1 are set to the same value.